For questions about recurrence relations, convergence tests, and identifying sequences

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20
votes
3answers
485 views

Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}$

Please help me to find a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
37
votes
4answers
781 views

Proving $\sum\limits_{l=1}^n \sum\limits _{k=1}^{n-1}\tan \frac {lk\pi }{2n+1}\tan \frac {l(k+1)\pi }{2n+1}=0$

Prove that $$\sum _{l=1}^{n}\sum _{k=1}^{n-1}\tan \frac {lk\pi } {2n+1}\tan \frac {l( k+1) \pi } {2n+1}=0$$ It is very easy to prove this identity for each fixed $n$ . For example let $n = 6$; ...
26
votes
4answers
668 views

A sum containing harmonic numbers $\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n}$

I'm trying to find a closed form for the following sum $$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$ where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number. Could you help me with it?
21
votes
4answers
629 views

Trying to prove that $\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$

How could one prove that: $$\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$$ This is about as far as I got: $$\prod_{k=1}^j \frac{2 k}{j+k-1} = \frac{2^j j!}{(j)_j} \implies$$ ...
17
votes
2answers
516 views

Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.

What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
12
votes
4answers
1k views

How to solve this sequence $165,195,255,285,345,x$

This is a question appeared in a competitive exam. The question is: Find the unknown term in $165,195,255,285,345,x$ 1)375 $\ \ \ \ \ \ \ \ $ 2)420 3)435 $\ \ \ \ \ \ \ ...
14
votes
2answers
580 views

How come such different methods result in the same number, $e$?

I guess the proof of the identity $$ \sum_{n = 0}^{\infty} \frac{1}{n!} \equiv \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x $$ explains the connection between such different calculations. How ...
19
votes
4answers
2k views

What's the value of $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$?

For some series, it is easy to say whether it is convergent or not by the "convergence test", e.g., ratio test. However, it is nontrivial to calculate the value of the sum when the series converges. ...
13
votes
4answers
729 views

Slowing down divergence 2

Let $f(x)$ and $g(x)$ be positive nondecreasing functions such that $ \sum_{n>1} \frac1{f(n)} \text{ and } \sum_{n>1} \frac1{g(n)} $ diverges. (Why) must the series $$\sum_{n>1} ...
26
votes
2answers
5k views

$\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this?

I came across a question where I needed to find the sum of the factorials of the first $n$ numbers. So I was wondering if there is any generic formula for this? Like there is a generic formula for ...
12
votes
4answers
589 views

Do these series converge to logarithms?

It is well known that $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}... =\log(2).$$ If we consider the array: $T(n,k) = -(n-1)\; \text{ if }\; n|k, \;\text{ else } \;1,$ Starting: ...
15
votes
1answer
701 views

Is there a constructive proof of this characterization of $\ell^2$?

I would like to revisit this question, which can be equivalently stated as: Proposition. Let $(a_n)$ be a sequence of real (or complex) numbers such that $\sum a_n b_n$ converges for every $(b_n) ...
10
votes
2answers
509 views

How to find $\zeta(0)=\frac{-1}{2}$ by definition?

I would like to know how we can find the following result: $$\zeta(0)=-\frac12$$ Is there a way, using the definition, $$\zeta(s)=\sum_{i=1}^{\infty}i^{-s}$$ to find this?
6
votes
3answers
539 views

Summing the series $(-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k+1}$

How does one sum the series, $$ S = a -\frac{2}{3}a^{3} + \frac{2 \cdot 4}{3 \cdot 5} a^{5} - \frac{ 2 \cdot 4 \cdot 6}{ 3 \cdot 5 \cdot 7}a^{7} + \cdots $$ This was asked to me by a high school ...
20
votes
3answers
1k views

Generalizing $\sum \limits_{n=1}^{\infty }n^{2}/x^{n}$ to $\sum \limits_{n=1}^{\infty }n^{p}/x^{n}$

For computing the present worth of an infinite sequence of equally spaced payments $(n^{2})$ I had the need to evaluate ...
14
votes
3answers
581 views

is a net stronger than a transfinite sequence for characterizing topology?

For metric spaces, knowledge of the convergence of sequences determines the topology completely. A set is closed in the metric topology if and only if it is closed under the limit of convergent ...
5
votes
3answers
329 views

Does $\sum _{k=2} ^\infty \frac{(-1)^k}{\sqrt{k}+(-1)^k}$ converge conditionally?

Discuss the convergence of the sum: $$\sum _{k=2} ^\infty \frac{(-1)^k}{\sqrt{k}+(-1)^k}.$$ My answer so far: It does not converge absolutely since $$\left | \frac{(-1)^k}{\sqrt{k}+(-1)^k} \right ...
3
votes
6answers
1k views

Bernoulli's representation of Euler's number, i.e $e=\lim \limits_{x\to \infty} \left(1+\frac{1}{x}\right)^x $ [duplicate]

Possible Duplicates: Finding the limit of $n/\sqrt[n]{n!}$ How come such different methods result in the same number, $e$? I've seen this formula several thousand times: $$e=\lim_{x\to ...
2
votes
4answers
335 views

Minimal solution to the sum using median

Suppose we have a set $S$ of real numbers. Show that $$\sum_{s\in S}|s-x| $$ is minimal if $x$ is equal to the median. This is a sample exam question of one of the exams that I need to take and I ...
6
votes
3answers
6k views

Summation involving a factorial: $1 + \sum_{j=1}^{n} j!j$ [duplicate]

$$1 + \sum_{j=1}^{n} j!j$$ I want to find a formula for the above and then prove it by induction. The answer according to Wolfram is $(n+1)!-1$, however I have no idea how to get there. Any hints or ...
7
votes
5answers
592 views

Proving $\sum_{k=1}^n k\cdot k! = (n+1)!-1$ without using mathematical Induction. [duplicate]

Possible Duplicate: Summation of a factorial This equation is given: $$ 1\cdot1! + 2\cdot2! + 3\cdot3! + \ldots + n\cdot n! = (n+1)! - 1 $$ I've solved it using mathematical induction but ...
5
votes
2answers
261 views

Convergence of $\sum_{n=2}^\infty \frac{1}{n^\alpha \ln^\beta (n)} $

Study the convergence of the following series: $$\sum_{n=2}^\infty \frac{1}{n^\alpha \cdot\ln^\beta(n)} \text{ where }\alpha,\beta \geq 0 $$ Applying d'Alembert criterion I have that $$ ...
3
votes
4answers
8k views

Is the sum of all natural numbers $-\frac{1}{12}$? [duplicate]

My friend showed me this youtube video in which the speakers present a line of reasoning as to why $$ \sum_{n=1}^\infty n = -\frac{1}{12} $$ My reasoning, however, tells me that the previous ...
5
votes
2answers
600 views

What's the limit of the sequence $\lim_{n \rightarrow \infty} \frac{n!}{n^n}$?

$\displaystyle \lim_{n \rightarrow \infty} \frac{n!}{n^n}$ I have a question: Is it valid to use Stirling's Formula to prove convergence of the sequence?
1
vote
2answers
821 views

Proving $\sum\limits_{i=0}^n i 2^{i-1} = (n+1) 2^n - 1$ by induction

I'm trying to apply an induction proof to show that $((n+1) 2^n - 1 $ is the sum of $(i 2^{i-1})$ from $0$ to $n$. the base case: L.H.S = R.H.S we assume that $(k+1) 2^k - 1 $ is true. we need to ...
-4
votes
2answers
587 views

How many pairs of integers $(A, B)$ are there in the range $[1,\ldots, N]$, such that $\gcd(A,B) = B$?

I am given a positive integer $N$ ($N\leq 10^9$). How many pairs of integers $(A, B)$ exist in the range $[1,\ldots, N]$ such that $\gcd(A,B) = B$?
79
votes
4answers
2k views

Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: ...
35
votes
3answers
1k views

Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge?

Does the following series converge or diverge? I would like to see a demonstration. $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}}. $$ I can see that: $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + ...
34
votes
5answers
1k views

A question on Taylor Series and polynomial

Suppose $ f(x)$ that is infinitely differentiable in $[a,b]$. For every $c\in[a,b] $ the series $\sum\limits_{n=0}^\infty \cfrac{f^{(n)}(c)}{n!}(x-c)^n $ is a polynomial. Is true that $f(x)$ is a ...
48
votes
4answers
4k views

Motivation for Ramanujan's mysterious $\pi$ formula

The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or ...
23
votes
3answers
453 views

$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2)$

How can I prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2).$$ Can anyone help me please?
18
votes
2answers
2k views

A hard definite integral with trigonometric functions

How could we get a closed form for this one? $$\displaystyle\int_{0}^{\frac{\pi }{2}}{{{x}^{2}}\sqrt{\tan x}\sin \left( 2x \right)\text{d}x}$$
31
votes
3answers
1k views

Prove the convergence/divergence of $\sum \limits_{k=1}^{\infty} \frac{\tan(k)}{k}$

Can be easily proved that the following series onverges/diverges? $$\sum_{k=1}^{\infty} \frac{\tan(k)}{k}$$ I'd really appreciate your support on this problem. I'm looking for some easy proof here. ...
22
votes
6answers
1k views

A closed form for the sum $\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$

How can I find a closed form for the following sum? $$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$ ($H_n=\sum_{k=1}^n\frac{1}{k}$).
28
votes
3answers
2k views

Why is this series of square root of twos equal $\pi$?

Wikipedia claims this but only cites an offline proof: $$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+... \sqrt 2}} = \pi$$ for $n$ square roots and one minus sign. The formula is not the "usual" one, like ...
42
votes
5answers
3k views

$n$th derivative of $e^{1/x}$

I am trying to find the $n$'th derivative of $f(x)=e^{1/x}$. When looking at the first few derivatives I noticed a pattern and eventually found the following formula $$\frac{\mathrm d^n}{\mathrm ...
19
votes
4answers
4k views

Convergence of the sequence $\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots$

I recently came across this problem Q1 Show that $\lim\limits_{n \rightarrow \infty} \underbrace{{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}}}_{n \textrm{ times }} = 3$ After ...
22
votes
2answers
830 views

Find the exact value of the infinite sum $\displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\!\Big)^{n}\right\}$

How can we find the exact value of the infinite sum $$ \displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\Big)^n\right\}? $$ This problem appears in: T. Andreescu, T. Radulescu & V. ...
19
votes
4answers
525 views

How close can $\sum_{k=1}^n \sqrt{k}$ be to an integer?

How close can $S(n) = \sum_{k=1}^n \sqrt{k}$ be to an integer? Is there some $f(n)$ such that, if $I(x)$ is the closest integer to $x$, then $|S(n)-I(S(n))|\ge f(n)$ (such as $1/n^2$, $e^{-n}$, ...). ...
17
votes
4answers
1k views

The sum of $(-1)^n \frac{\ln n}{n}$

I'm stuck trying to show that $$\sum_{n=2}^{\infty} (-1)^n \frac{\ln n}{n}=\gamma \ln 2- \frac{1}{2}(\ln 2)^2$$ This is a problem in Calculus by Simmons. It's in the end of chapter review and it's ...
16
votes
2answers
940 views

Evaluating $\sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1}$ using the inverse Mellin transform

Inspired by this post, I'm trying to evaluate $\displaystyle \sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1} = \frac{1}{24} - \frac{1}{8 \pi}$ using the inverse Mellin transform. But my answer is twice ...
12
votes
4answers
924 views

Testing the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{k + \cos{n}}}$

We know that the "Harmonic Series" $$ \sum \frac{1}{n}$$ diverges. And for $p >1$ we have the result that the series converges $$\sum \frac{1}{n^{p}}$$ converges. One can then ask the question of ...
8
votes
5answers
1k views

Prove $\lim_{x \rightarrow 0} \frac {\sin(x)}{x} = 1$ with the epsilon-delta definition of limit.

It is well known that $$\lim_{x \rightarrow 0} \frac {\sin(x)}{x} = 1$$ I know several proofs of this: the geometric proof shows that $\cos(\theta)\leq\frac {\sin(\theta)}{\theta}\leq1$ and using ...
9
votes
2answers
1k views

Does $\sum \limits_{n=1}^\infty\frac{\sin n}{n}(1+\frac{1}{2}+\cdots+\frac{1}{n})$ converge (absolutely)?

I've had no luck with this one. None of the convergence tests pop into mind. I tried looking at it in this form $\sum \sin n\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}$ and apply Dirichlets test. I ...
8
votes
5answers
370 views

Finding sum of a series: difference of cubes

I am trying to find sum of the infinite series: $$1^{3}-2^{3}+3^{3}-4^{3}+5^{3}-6^{3} + \ldots$$ I tried to solve it by subtracting sum of even cubes from odd, but that solves only half of the ...
6
votes
3answers
364 views

sum of arctangent

Here is an interesting topic. It comes from evaluating $$\sum_{k=1}^{\infty}tan^{-1}\left(\frac{1}{k^{2}}\right)$$ I managed to dig up an old paper I have on the sum of arctans by Boros and Moll. ...
13
votes
2answers
2k views

Series of logarithms $\sum\limits_{k=1}^\infty \ln(k)$ (Ramanujan summation?)

I had this question earlier, so to say as a "standalone" problem, but now it pops up in context of an analysis with the lngamma-function. As well as we can convert the question of sums of like powers ...
12
votes
2answers
590 views

Double sum trouble

Evaluate: $$\sum_{j=1}^{\infty} \sum_{i=1}^{\infty} \frac{j^2i}{3^j(j3^i+i3^j)}$$ Honestly, I don't see where to start with this. I am sure that this is a trick question and I am missing something ...
8
votes
4answers
629 views

Find the limit $ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$

For $a,b,c>0$, Find $$ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$$ how can I find the limit of sequence above? Provide me a hint or full solution. thanks ^^
10
votes
2answers
2k views

Determining precisely where $\sum_{n=1}^\infty\frac{z^n}{n}$ converges?

Inspired by the exponential series, I'm curious about where exactly the series $\displaystyle\sum_{n=1}^\infty\frac{z^n}{n}$ for $z\in\mathbb{C}$ converges. I calculated $$ ...